Stage-parallel fully implicit Runge–Kutta solvers for discontinuous Galerkin fluid simulations
Abstract
In this paper, we develop new techniques for solving the large, coupled linear systems that arise from fully implicit Runge–Kutta methods. This method makes use of the iterative preconditioned GMRES algorithm for solving the linear systems, which has seen success for fluid flow problems and discontinuous Galerkin discretizations. By transforming the resulting linear system of equations, one can obtain a method which is much less computationally expensive than the untransformed formulation, and which compares competitively with other time-integration schemes, such as diagonally implicit Runge–Kutta (DIRK) methods. We develop and test several ILU-based preconditioners effective for these large systems. We additionally employ a parallel-in-time strategy to compute the Runge–Kutta stages simultaneously. Numerical experiments are performed on the Navier–Stokes equations using Euler vortex and 2D and 3D NACA airfoil test cases in serial and in parallel settings. In conclusion, the fully implicit Radau IIA Runge–Kutta methods compare favorably with equal-order DIRK methods in terms of accuracy, number of GMRES iterations, number of matrix–vector multiplications, and wall-clock time, for a wide range of time steps.
- Authors:
-
[1];
- Brown Univ., Providence, RI (United States)
- Univ. of California, Berkeley, CA (United States)
- Publication Date:
- Research Org.:
- Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States). National Energy Research Scientific Computing Center (NERSC); Univ. of California, Oakland, CA (United States)
- Sponsoring Org.:
- USDOE
- OSTI Identifier:
- 1543560
- Alternate Identifier(s):
- OSTI ID: 1398120
- Grant/Contract Number:
- AC02-05CH11231
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Journal of Computational Physics
- Additional Journal Information:
- Journal Volume: 335; Journal Issue: C; Journal ID: ISSN 0021-9991
- Publisher:
- Elsevier
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Computer Science; Physics; Implicit Runge–Kutta; Discontinuous Galerkin; Preconditioned GMRES; Parallel-in-time
Citation Formats
Pazner, Will, and Persson, Per -Olof. Stage-parallel fully implicit Runge–Kutta solvers for discontinuous Galerkin fluid simulations. United States: N. p., 2017.
Web. doi:10.1016/j.jcp.2017.01.050.
Pazner, Will, & Persson, Per -Olof. Stage-parallel fully implicit Runge–Kutta solvers for discontinuous Galerkin fluid simulations. United States. https://doi.org/10.1016/j.jcp.2017.01.050
Pazner, Will, and Persson, Per -Olof. Fri .
"Stage-parallel fully implicit Runge–Kutta solvers for discontinuous Galerkin fluid simulations". United States. https://doi.org/10.1016/j.jcp.2017.01.050. https://www.osti.gov/servlets/purl/1543560.
@article{osti_1543560,
title = {Stage-parallel fully implicit Runge–Kutta solvers for discontinuous Galerkin fluid simulations},
author = {Pazner, Will and Persson, Per -Olof},
abstractNote = {In this paper, we develop new techniques for solving the large, coupled linear systems that arise from fully implicit Runge–Kutta methods. This method makes use of the iterative preconditioned GMRES algorithm for solving the linear systems, which has seen success for fluid flow problems and discontinuous Galerkin discretizations. By transforming the resulting linear system of equations, one can obtain a method which is much less computationally expensive than the untransformed formulation, and which compares competitively with other time-integration schemes, such as diagonally implicit Runge–Kutta (DIRK) methods. We develop and test several ILU-based preconditioners effective for these large systems. We additionally employ a parallel-in-time strategy to compute the Runge–Kutta stages simultaneously. Numerical experiments are performed on the Navier–Stokes equations using Euler vortex and 2D and 3D NACA airfoil test cases in serial and in parallel settings. In conclusion, the fully implicit Radau IIA Runge–Kutta methods compare favorably with equal-order DIRK methods in terms of accuracy, number of GMRES iterations, number of matrix–vector multiplications, and wall-clock time, for a wide range of time steps.},
doi = {10.1016/j.jcp.2017.01.050},
journal = {Journal of Computational Physics},
number = C,
volume = 335,
place = {United States},
year = {Fri Jan 27 00:00:00 EST 2017},
month = {Fri Jan 27 00:00:00 EST 2017}
}
Free Publicly Available Full Text
Publisher's Version of Record
Other availability
Search WorldCat to find libraries that may hold this journal
Cited by: 32 works
Citation information provided by
Web of Science
Web of Science
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.
Works referenced in this record:
Diagonally Implicit Runge–Kutta Methods for Stiff O.D.E.’s
journal, December 1977
- Alexander, Roger
- SIAM Journal on Numerical Analysis, Vol. 14, Issue 6
Linearly implicit Rosenbrock-type Runge–Kutta schemes applied to the Discontinuous Galerkin solution of compressible and incompressible unsteady flows
journal, September 2015
- Bassi, F.; Botti, L.; Colombo, A.
- Computers & Fluids, Vol. 118
Implicit Time Integration Schemes for the Unsteady Compressible Navier–Stokes Equations: Laminar Flow
journal, June 2002
- Bijl, Hester; Carpenter, Mark H.; Vatsa, Veer N.
- Journal of Computational Physics, Vol. 179, Issue 1
On the implementation of implicit Runge-Kutta methods
journal, September 1976
- Butcher, J. C.
- BIT, Vol. 16, Issue 3
Fourth-Order Runge–Kutta Schemes for Fluid Mechanics Applications
journal, October 2005
- Carpenter, M. H.; Kennedy, C. A.; Bijl, Hester
- Journal of Scientific Computing, Vol. 25, Issue 1
Order Results for Implicit Runge–Kutta Methods Applied to Stiff Systems
journal, June 1985
- Frank, Reinhard; Schneid, Josef; Ueberhuber, Christoph W.
- SIAM Journal on Numerical Analysis, Vol. 22, Issue 3
Validation of a High-Order Large-Eddy Simulation Solver Using a Vertical-Axis Wind Turbine
journal, January 2016
- Kanner, Samuel; Persson, Per-Olof
- AIAA Journal, Vol. 54, Issue 1
Singly Diagonally Implicit Runge–Kutta Methods with an Explicit First Stage
journal, August 2004
- Kværnø, A.
- BIT Numerical Mathematics, Vol. 44, Issue 3
An incomplete factorization technique for positive definite linear systems
journal, May 1980
- Manteuffel, T. A.
- Mathematics of Computation, Vol. 34, Issue 150
Modified extended BDF scheme for the discontinuous Galerkin solution of unsteady compressible flows: DG-MEBDF UNSTEADY SOLUTION OF COMPRESSIBLE FLOWS
journal, August 2014
- Nigro, A.; Ghidoni, A.; Rebay, S.
- International Journal for Numerical Methods in Fluids, Vol. 76, Issue 9
Up to sixth-order accurate A-stable implicit schemes applied to the Discontinuous Galerkin discretized Navier–Stokes equations
journal, November 2014
- Nigro, Alessandra; De Bartolo, Carmine; Bassi, Francesco
- Journal of Computational Physics, Vol. 276
The Compact Discontinuous Galerkin (CDG) Method for Elliptic Problems
journal, January 2008
- Peraire, J.; Persson, P. -O.
- SIAM Journal on Scientific Computing, Vol. 30, Issue 4
Newton-GMRES Preconditioning for Discontinuous Galerkin Discretizations of the Navier–Stokes Equations
journal, January 2008
- Persson, P. -O.; Peraire, J.
- SIAM Journal on Scientific Computing, Vol. 30, Issue 6
High-order CFD methods: current status and perspective: HIGH-ORDER CFD METHODS
journal, January 2013
- Wang, Z. J.; Fidkowski, Krzysztof; Abgrall, Rémi
- International Journal for Numerical Methods in Fluids, Vol. 72, Issue 8
A special stability problem for linear multistep methods
journal, March 1963
- Dahlquist, Germund G.
- BIT, Vol. 3, Issue 1
Linearly implicit Rosenbrock-type Runge–Kutta schemes applied to the Discontinuous Galerkin solution of compressible and incompressible unsteady flows
journal, September 2015
- Bassi, F.; Botti, L.; Colombo, A.
- Computers & Fluids, Vol. 118
A parallelizable preconditioner for the iterative solution of implicit Runge–Kutta-type methods
journal, November 1999
- Jay, Laurent O.; Braconnier, Thierry
- Journal of Computational and Applied Mathematics, Vol. 111, Issue 1-2
High-Order Discontinuous Galerkin Methods for CFD
book, March 2011
- Peraire, Jaime; Persson, Per-Olof
- Advances in Computational Fluid Dynamics
Works referencing / citing this record:
Evaluation of Fully Implicit Runge Kutta Schemes for Unsteady Flow Calculations
journal, July 2017
- Jameson, Antony
- Journal of Scientific Computing, Vol. 73, Issue 2-3
Analysis and Entropy Stability of the Line-Based Discontinuous Galerkin Method
journal, March 2019
- Pazner, Will; Persson, Per-Olof
- Journal of Scientific Computing, Vol. 80, Issue 1
Analysis and entropy stability of the line-based discontinuous Galerkin method
text, January 2018
- Pazner, Will; Persson, Per-Olof
- arXiv
Analysis and Entropy Stability of the Line-Based Discontinuous Galerkin Method
journal, March 2019
- Pazner, Will; Persson, Per-Olof
- Journal of Scientific Computing, Vol. 80, Issue 1
Reinterpretation of Multi-Stage Methods for Stiff Systems: A Comprehensive Review on Current Perspectives and Recommendations
journal, December 2019
- Jeon, Yonghyeon; Bak, Soyoon; Bu, Sunyoung
- Mathematics, Vol. 7, Issue 12
Irksome: Automating Runge--Kutta time-stepping for finite element methods
preprint, January 2020
- Farrell, Patrick E.; Kirby, Robert C.; Marchena-Menendez, Jorge
- arXiv
Kernel-based Active Subspaces with application to CFD parametric problems using Discontinuous Galerkin method
text, January 2020
- Romor, Francesco; Tezzele, Marco; Lario, Andrea
- arXiv